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Channel Estimation for RIS-Aided Multiuser Millimeter-Wave Systems

Gui Zhou, Cunhua Pan, Hong Ren, Petar Popovski, A. Lee Swindlehurst

arXiv:2106.14792v2eess.SP

TL;DR

Passive RIS hardware and many reflecting elements make cascaded CSI estimation costly. The paper exploits mmWave sparsity, cascaded-path correlation, shared BS-RIS structure, and slowly varying angles through a two-phase first-block protocol followed by gain-only updates. It reports substantially lower pilot overhead and improved NMSE relative to existing OMP-based methods.

  • Problem

    Passive RIS elements make CSI acquisition challenging, while large RISs create high channel-estimation overhead.

  • Method

    The method estimates full CSI initially, then re-estimates only gains using persistent angle information, path correlation, common BS-RIS CSI, and optimized RIS training phase shifts.

  • Results

    The proposed algorithm outperforms existing OMP-based methods in NMSE, pilot overhead, and computational complexity, with NMSE close to the low-SNR lower bound.

  • Takeaways & Limitations

    The protocol provides a low-overhead cascaded-channel estimation strategy for RIS-aided multiuser mmWave systems across consecutive coherence blocks.

Abstract

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Channel estimation in the RIS-aided massive multiuser multiple-input single-output (MU-MISO) wireless communication systems is challenging due to the passive feature of RIS and the large number of reflecting elements that incur high channel estimation overhead. To address this issue, we propose a novel cascaded channel estimation strategy with low pilot overhead by exploiting the sparsity and the correlation of multiuser cascaded channels in millimeter-wave massive MISO systems. Based on the fact that the phsical positions of the BS, the RIS and users may not change in several or even tens of consecutive channel coherence blocks, we first estimate the full channel state information (CSI) including all the angle and gain information in the first coherence block, and then only re-estimate the channel gains in the remaining coherence blocks with much less pilot overhead. In the first coherence block, we propose a two-phase channel estimation method, in which the cascaded channel of one typical user is estimated in Phase I based on the linear correlation among cascaded paths, while the cascaded channels of other users are estimated in Phase II by utilizing the partial CSI of the common base station (BS)-RIS channel obtained in Phase I. The total theoretical minimum pilot overhead in the first coherence block is $8J-2+(K-1)\left\lceil (8J-2)/L\right\rceil $, where $K$, $L$ and $J$ denote the numbers of users, paths in the BS-RIS channel and paths in the RIS-user channel, respectively. In each of the remaining coherence blocks, the minimum pilot overhead is $JK$. Moreover, the training phase shift matrices at the RIS are optimized to improve the estimation performance.

I. INTRODUCTION

RISs can improve wireless coverage and capacity, but passive reflecting elements make CSI acquisition difficult and conventional estimation incurs high pilot overhead. This paper exploits mmWave sparsity, cascaded-path correlation, common BS-RIS parameters, and slowly varying angles to reduce overhead while estimating unknown sparsity levels.

  • Passive RIS elements cannot transmit, receive, or process pilot signals, making accurate cascaded CSI acquisition challenging.
  • LS-based estimation requires at least M pilots per user, which is impractical for RISs with many reflecting elements.
  • Existing low-overhead methods exploit common multiuser parameters or mmWave sparsity, but face rank-deficiency, power leakage, false alarms, or known-path assumptions.
  • The proposed protocol estimates full CSI in the first coherence block and only channel gains thereafter, requiring JK pilots in later blocks.
  • In the first block, Phase I exploits linear cascaded-path correlation with overhead 8J − 2, while Phase II uses common BS-RIS CSI and requires (K−1)⌈(8J −2)/L⌉ pilots.
  • Simulations report lower MSE, pilot overhead, and computational complexity than OMP-based estimation, with performance near the low-SNR lower bound.

II. SYSTEM AND CHANNEL MODEL

The paper models uplink cascaded-channel estimation in a narrow-band TDD mmWave massive MISO system with blocked direct BS-user links. Users transmit pilots sequentially while RIS phase shifts configure the reflected links observed at the BS.

  • The system contains K single-antenna users, an N-antenna BS, and an RIS with M passive reflecting elements.
  • Because direct BS-user channels are blocked, estimation focuses on the user-RIS and RIS-BS uplink links.
  • The RIS applies a unit-modulus phase-shift vector e_t at each time slot, while users transmit pilot sequences one at a time.
  • The received pilot measurements at the BS include the cascaded channel contribution and additive white Gaussian noise with power δ^2.
  • The work estimates cascaded user-RIS-BS channels because joint active BS and passive RIS beamforming depends on them.
  • An unbiased LS estimator can require τ_k ≥ M pilots per user, motivating sparse mmWave-based alternatives.

B. Cascaded Channel Sparsity Model

The cascaded mmWave channel is sparse and structurally correlated: its JL cascaded paths arise from only J+L independent spatial paths, while all users share the BS-RIS channel. This structure reduces the effective parameters needed for estimation.

  • For user k, the BS-RIS and RIS-user links contain L and J_k propagation paths with complex gains α_l and β_k,j, respectively.
  • The cascaded path uses a RIS steering vector indexed by the difference between BS-RIS and user-RIS directional cosines.
  • The geometric channel model exhibits low rank and spatial correlation, motivating compressed-sensing estimation in RIS-aided mmWave systems.
  • The angular-domain cascaded matrix is sparse, but prior methods estimate L AoAs, J_kL cascaded AoD cosines, and J_kL complex gains.
  • Only J_k+L complex gains and 2L+J_k angles need estimation per user, and users share the same L BS-RIS gains.

III. CHANNEL ESTIMATION

The protocol estimates full CSI in the first coherence block, then re-estimates only channel gains when angles remain unchanged across blocks. It exploits massive-array sparsity and angle rotation to estimate common AoAs accurately.

  • Channel Estimation Protocol: The protocol estimates full CSI, including angles and gains, in the first coherence block and only channel gains afterward.This assumes BS, RIS, and users maintain positions sufficiently for angles to remain unchanged while gains vary between blocks.
  • Channel Estimation Protocol: User 1 sends pilots to estimate AoAs, cascaded AoD cosines, and gains, enabling construction of a reparameterized common BS-RIS channel for other users.The remaining users then transmit pilot symbols for their channel estimation.
  • Estimation of the common AoAs: The common BS-RIS channel is estimated from row sparsity in the DFT-domain measurement matrix, whose nonzero rows correspond to common AoAs.The method identifies peak-power rows and estimates the number of propagation paths from the detected nonzero rows.
  • Estimation of the common AoAs: Angle rotation compensates for DFT angle-grid mismatch and concentrates channel power on the target row, improving AoA estimation accuracy.Finite DFT resolution causes mismatch between discrete and continuous angles; one-dimensional search selects the rotation parameter.
  • Estimation of the common AoAs: In the example with N = M = 100 and L = 1, optimal rotation focuses more power at φ = 14°, whereas leakage spreads the unrotated beam.The power peak remains useful for initial AoA estimation, while rotation makes the estimate more accurate.

2) Estimation of the cascaded AoD cosines and gains:

For one typical user, the method estimates cascaded paths by combining sparse recovery with an angle-gain scaling relation among paths. This avoids independently recovering every cascaded path.

  • 2) Estimation of the cascaded AoD cosines and gains:: The measurement model is transformed into a sparse signal recovery problem using an overcomplete steering-vector dictionary, solvable with compressed-sensing methods such as OMP.The sparse vector contains J1 nonzero cascaded gains, while the RIS phase-shift matrix is designed for better estimation.
  • 2) Estimation of the cascaded AoD cosines and gains:: 8J1 − 2 measurements are sufficient to recover a J1-sparse complex-valued signal vector.This lower bound is used to determine the pilot requirement for the reference-path estimation.
  • 2) Estimation of the cascaded AoD cosines and gains:: The angle-gain scaling property represents every h_RIS,l using one arbitrary reference path h_RIS,r and its relative angle and gain parameters.For each non-reference path, the method estimates Δω_l and x_l, then reconstructs h_RIS,l from the reference estimate.
  • 2) Estimation of the cascaded AoD cosines and gains:: The estimated cascaded AoD cosines and gains from the reference path facilitate cascaded-channel estimation for the other users.The complete estimate is formed from the common steering matrix and the estimated RIS-side cascaded channels.
  • 2) Estimation of the cascaded AoD cosines and gains:: The algorithm uses τ1 ≥ 8J1 − 2 time slots to estimate 3L + 2J1 − 2 parameters and recover G1 of dimension N × M.The required number of time slots is not related to L.

C. Channel Estimation for Other Users in the First Coherence Block

For users beyond the typical user, the method constructs a reparameterized common BS-RIS channel from the first user’s estimate and uses it to recover each remaining user’s sparse CSI with reduced overhead.

  • C. Channel Estimation for Other Users in the First Coherence Block: All users share common channel matrices derived from the estimated common AoA steering matrix, enabling construction of the reparameterized common channel Hc.The estimate is formed as bHc = bAN bΛc bAH.
  • C. Channel Estimation for Other Users in the First Coherence Block: The original BS-RIS channel cannot be directly recovered from user 1’s cascaded channel because angles and gains are coupled within each cascaded subpath.The method instead constructs a substitute Hc containing reparameterized information about the common channel.
  • C. Channel Estimation for Other Users in the First Coherence Block: For each remaining user, the projected and vectorized measurements are approximated as sparse recovery problems and solved using compressed-sensing methods such as OMP.The sparse vector ck contains Jk gains, and the equivalent dictionary is constructed from Hc-related quantities.
  • C. Channel Estimation for Other Users in the First Coherence Block: Algorithm 3 constructs the equivalent dictionary from the estimated common-channel parameters and outputs the cascaded-channel estimate bGk for users 2 ≤ k ≤ K.The algorithm returns the common steering matrix, constructs bΛc and bAc, and applies sparse recovery.
  • C. Channel Estimation for Other Users in the First Coherence Block: The pilot overhead for user k satisfies τk ≥ (8Jk − 2)/L.This follows from requiring τkL ≥ 8Jk − 2 measurements to recover the Jk-sparse signal.

D. Channel Estimation in the Remaining Coherence Blocks

In subsequent coherence blocks, the method reuses angle information estimated initially and estimates only changing cascaded gains using least squares with shorter pilots.

  • D. Channel Estimation in the Remaining Coherence Blocks: With angle information from the first coherence block, only cascaded channel gains need to be re-estimated in later blocks.The updated gains and previously estimated angles are then used to reconstruct each user’s uplink channel.
  • D. Channel Estimation in the Remaining Coherence Blocks: The remaining-block measurement model uses previously estimated steering matrices B̂k,l and an LS estimate of the changing channel gains.The steering matrices are obtained from first-block angle estimates for user 1 and the other users.
  • D. Channel Estimation in the Remaining Coherence Blocks: The pilot length must satisfy τk ≥ Jk for user k because the pilot-related matrix must support the required pseudo-inverse operation.This provides the per-user pilot requirement for gain re-estimation in subsequent blocks.

IV. TRAINING REFLECTION COEFFICIENT OPTIMIZATION

The section optimizes RIS training phase-shift matrices by targeting incoherent equivalent dictionaries, improving OMP-based recovery of sparse cascaded-channel signals under unit-modulus constraints.

  • OMP recovery performance is positively related to the orthogonality of the equivalent dictionary.
  • The training phase-shift matrices are optimized to generate approximately orthogonal equivalent dictionaries.The designs target recovery of the sparse signals b_l and c_k in the channel-estimation models.
  • The proposed design extends constrained phase-shift optimization beyond prior unconstrained and constrained solutions.The section presents a more concise solution based on earlier approaches.
  • The optimization begins by reducing the design problem and decomposing AAH into eigenvalues and eigenvectors.A matrix Γ with orthogonal rows is then constructed, for example as [Iτ1 0].

E1 ||EH

For E1, the constrained design is transformed into quadratic and least-squares subproblems, then solved by alternating optimization of the training matrix and auxiliary matrix Γ.

  • An alternating-optimization method designs Γ and E_k, with Γ construction accounting for the structure of Z_k.For k ≥ 2, a pre-designed E_k derived from a DFT matrix is used while Γ is constructed through an orthogonal Procrustes problem.
  • The complicated structure of the E_k design is reconstructed through mathematical transformations so E_k can be written in quadratic form.
  • The transformed problem is further reformulated using parallel stacking and solved as a least-squares problem.
  • The final E_k solution enforces the unit-modulus constraint after obtaining the unconstrained least-squares solution.
  • The two optimization problems are alternately optimized until a stopping criterion is satisfied.

V. ANALYSIS OF PILOT OVERHEAD AND COMPUTATIONAL COMPLEXITY

The proposed strategy reduces pilot overhead and estimation complexity by exploiting channel sparsity, shared structure, and persistent angle information across coherence blocks. Simulations show improved NMSE with low overhead under the evaluated settings.

  • Pilot overhead: 8J-2+(K-1)ceil((8J-2)/L) is the first-coherence-block pilot overhead, while subsequent blocks require JK pilots.The first block estimates full CSI; later blocks update only cascaded channel gains.
  • Computational complexity: O(Ng + 8DJ + K8DJ^4) is the total first-block estimation complexity, while later blocks use LS updates with complexity on the order of O(J^3).The proposed method's complexity is much lower under L ≪ N(M), J ≪ N(M), and g ≪ N.
  • Pilot-overhead simulations: T = 14 time slots for Proposed-full-CSI outperforms LS with T = M = 100 time slots at SNR=0 dB.The proposed estimator exploits the low-rank mmWave channel, whereas LS ignores channel sparsity.
  • Pilot-overhead simulations: T = 2J = 8 time slots for Proposed-gains surpasses LS after angle information is estimated.The strategy first estimates angles and then re-estimates only cascaded gains.
  • SNR and antenna simulations: At low SNR, the proposed algorithms outperform LS, while their estimation accuracy increases with SNR before saturating at relatively high SNR.The reported error floor is attributed to finite-N steering-matrix non-orthogonality and mismatch.
  • SNR and antenna simulations: When N exceeds 80, Proposed-full-CSI works well, whereas LS remains stable and the OMP-based benchmark performs poorly because of power leakage.Increasing pilot overhead from T=J to T=4J improves Proposed-gains performance through greater measurement-data diversity.

VII. CONCLUSIONS

The paper develops a low-overhead cascaded channel estimation method for RIS-aided multiuser mmWave systems by exploiting channel structure across users and coherence blocks. It characterizes minimum pilot overhead, designs training reflection matrices, and reports improved NMSE with substantially lower overhead than existing OMP-based methods.

  • The proposed method targets RIS-aided uplink multiuser mmWave systems with much less pilot overhead.
  • Angle information remains essentially static across many coherence blocks, enabling channel-gain re-estimation while exploiting linear correlation among cascaded paths.
  • The method uses reparameterized CSI of the common BS-RIS channel to support multiuser cascaded channel estimation.
  • Theoretical minimum pilot overhead is characterized, and training reflection matrices are designed for the estimation procedure.
  • Simulation results show that the proposed algorithm outperforms existing OMP-based algorithms in NMSE while requiring much less pilot overhead.

APPENDIX A

The appendix establishes asymptotic properties of array responses and their transformed representations. It uses these properties to show concentration of matrix powers at selected points and derive the stated sparse-matrix relation.

  • THE PROOF OF LEMMA 1: For distinct angles, the normalized inner product of array responses is bounded and vanishes asymptotically, while matching angles yield normalized inner product one.
  • THE PROOF OF LEMMA 1: The proof identifies the Dirac delta function in the limiting expression for the array-response inner products.
  • THE PROOF OF LEMMA 2: Asymptotic limits guarantee integer indices at which transformed array-response powers are concentrated.
  • THE PROOF OF LEMMA 2: The transformed matrix is sparse, with its powers concentrated at points indexed by the selected integers for each path.
  • THE PROOF OF LEMMA 2: The appendix derives equation (17) by combining equations (74) and (76).
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